I am trying to give an analogy: In exploring for mineral deposits, we compare different prospecting areas and look for those samples which yield economically viable grades (of minerals). Poisson distribution is used, which only considers those sample points which give acceptable grade. From these observations, an estimate is made using Poisson distribution, and the likelihood of occurrence of deposit in the remaining field is computed.

You are looking at the variable "rare behaviour". When you conduct a study and find group of animals exhibiting "rare behaviour" , you have an occurrence. In the entire target population, how many such occurrences are recorded by you?

In another group of animals, same recording/ observation is done (freeze the number of observations in both at the same level).

Now, based on these observations, using Poisson distribution, estimate the occurrence of "rare behaviour" in different groups of animals for n+m samples.

In majority of the cases, the research you conduct may be unique. You may not find much in the published literature, or whatever is published may not be applicable. Next step is to make use of analogous cases in other branches of science or engineering.

Groups like "Research Gate" give us an opportunity to share ideas or applications or insights.

Although these are older papers, they (or their reference lists) might be of some use, from a comparative perspective, to addressing your analytic problem:

Robert M. Fagen, Robert N. Goldman, 1977. Behavioural catalogue analysis methods. Animal Behaviour 25(2): 261-274.

Robert M. Fagen, 1978. Behavioural catalogue acquisition and completeness: Validation of logarithmic regression and curve-fitting methods. Animal Behaviour 26(3): 961-962.

Zero-inflated Poisson distributions are particularly difficult if the zero fraction changes e.g. from animal to animal, and small expected value . In particular, it can require a very large number of samples to determine accurately whether it is the zero or the expected value that is changing. If these are rare events, you probably don't have a very large sample size.

If the zero-inflated Poisson statistical tests provided in e.g. R are not good enough on their own, you can always fall back on alternate more general methods like bootstrap analysis or some sort of Bayesian estimate. You may need to contact a statistician to get the details right (though basic bootstrap is pretty hard to mess up, as long as you remember that your sampling has to match the structure of the data, e.g. if you run mulitple animals under one condition and get a count of behaviors per animal, you can't just put all the behaviors from those animals into one pool; you need to model the process of first selecting an animal and then selecting multiple responses from that animal).

I am not familiar with behavioral analyses, but if you are interested in comparing different samples of distributions, you may be interested in performing some functional data analyses (FDA; see for example Ramsay and Silverman, 2002; 2005). I brief, this family of statistical tests alows you to analyze samples were each individual is represented by a function (ie. curve or surface) that best describes the observed data. In this context, analyzing the shape of several distributions may be performed by analyzing, to give you an example, by using kernel density functions that represent each individual distribution.

Although these statistical tools may be quite complex to understand at the begining, I think that it is worth for you to take a look on this methodology and, after that, you decide if it serves you or not.

Good luck with your analyses!

Thanks all; these are giving us some directions to pursue. Let me clarify in case that spurs additional responses. We are analyziing 10 minute focal animal samples in gibbons across about 23 family groupings in about 6 locations. The number and distribution over the day of samples across animals within a location are standard but between locations are not. The rare events are behaviours such as overt agonism, copulation, etc.

We are analyzing behavioral profiles of African bovids, using 3-min videos. We use a point sampling technique for activity states and span sampling between the points for all occurrences of events. This provides a list of the diversity of behavioral events, which can be compared with an index (e.g. Shannon Weaver species diversity), as well as a time register if we want to go back in and look at the details of rare events. Counting the number of spans for which a particular event was recorded does not give a true frequency. However, it does give an index that can be compared across individuals using a non-parametric statistic such as log likelihood ratio test (G-square).

Your elaboration about analyzing focal animal data from multiple gibbon groups got me thinking further about the group comparison and rare behaviour problems. I'm also a primatologist, and was at a conference earlier this month where a graduate student presented a paper looking at preferred and fall-back foods in white-faced capuchin monkeys (also using 10-minute focal animal follows, wortking with four social groups). They used radial plots to visually characterize the data on availability of food types (fruit, catepillars, and non-caterpillar insect, expressed in terms of abundance by month), from which they recognized three seasons of food availability: high fruit, high caterpillar; high fruit. low caterpillar; and, low fruit, low caterpillar. They then compared mean foraging rates of the three food categories for each of the "food availability seasons" using Generalized Least Squares analysis. Perhaps your gibbon behavioural data could be usefully conceptualized through the use of radial plots, and from that suitable analyses conducted. I've poked around a little bit and found a couple of papers on analytical approaches that utilize radial plots (which are attached) -- hope this might be a productive line of inquiry!

A second paper on "Some Applications of Radial Plots" (Galbraith, 1994).

Thanks all for suggestions and references. Great starts. Researchgate is providing a great and quick forum for substantive interchange.

Use contingency analysis based on log-linear models (G-square and binomial z scores) to test the predicted association: "given that behavior X was observed, was it equally likely in groups V and W".

You can take a look to Zuur (2009) Mixed model in R. You can tackle the problem by using a GLMM! At first it seems quite confusing, but once you get the grip of it, it will help you to solve many problems. There is a chapter dealing zero inflated Poisson distribution.